For the bilinear control system of the form x = (A + Sigma(i=1)(m) u(i)B(i))x, x epsilon R(n), u(i) epsilon R, with A essentially nonnegative and Bi constant diagonal matrices, the following global controllability question is studied : when can any two points in R(n) with positive coordinates be connected by a trajectory of this system? The answers for m = n - 1 and m = n - 2 for any n > 2 are given; some necessary conditions for other cases are proven.